Mathematics

R. N. Abdulaev

On the Solvability Condition for the Homogeneous Riemann Problem on Closed Riemann Surfaces

(Presented by Academician I. N. Vekua on 23 V 1963)

1. Let on a closed Riemann surface \(R\) of genus \(\rho\) there be given a contour \(\Gamma\), consisting of a finite number of mutually nonintersecting smooth closed curves
\[ \Gamma=\bigcup_{j=1}^{m}\Gamma_j . \]

Consider the Riemann boundary-value problem
\[ \Phi^{+}(P)=G(P)\Phi^{-}(P) \tag{1} \]
under the assumption that \(G(P)\in H(\Gamma)\), \(G(P)\ne 0\).

The question of the solvability of problem (1) depending on the value of the index
\[ \varkappa=\frac{1}{2\pi i}\int_{\Gamma} d\ln G =\sum_{j=1}^{m}\frac{1}{2\pi i}\int_{\Gamma_j} d\ln G =\sum_{j=1}^{m}\varkappa_j \tag{2} \]
was studied in works \((^{4-6})\), where it was proved that for \(\varkappa\ge \rho\) the problem is always solvable (a regular solution is meant, not identically equal to zero), while for \(\varkappa<0\) there are no regular solutions except \(\Phi\equiv 0\).

In the present paper we give a necessary and sufficient condition for the solvability of problem (1) when \(0\le \varkappa<\rho\), in terms of Riemann \(\vartheta\)-functions, supplementing the results known for this case from work \((^{5})\).

2. Choose on each contour \(\Gamma_j\) one point \(P_j\) \((j=1,2,\ldots,m)\) as the initial point for traversing the contour, and an arbitrary branch \(\ln G\), single-valued on the contour \(\Gamma\) cut at the points \(P_j\) \((j=1,2,\ldots,m)\). Consider the function
\[ \Psi(z)=\exp\left\{\frac{1}{2\pi i}\int_{\Gamma}\ln G(t)\,A^{*}(t,z)\,dt\right\}, \tag{3} \]
where \(A^{*}(\zeta,z)\) is the multivalued Cauchy kernel with periods
\[ \int_{A_\nu} d_z A^{*}(\zeta,z)=0,\qquad \int_{B_\nu} d_z A^{*}(\zeta,z)=2\pi i\,\frac{dw_\nu}{d\zeta} \quad(\nu=1,2,\ldots,\rho), \tag{4} \]
and \(dw_1,\ldots,dw_\rho\) is a complex-normalized basis of Abelian differentials of the first kind \((^{7})\). From (4) and the known properties of an integral of Cauchy type \((^{1})\) it follows that \(\Psi(z)\) is a multiplicatively multivalued function with multipliers
\[ \mu(A_\nu)=1,\qquad \mu(B_\nu)=\exp\int_{\Gamma}\ln G\,dw_\nu \quad(\nu=1,\ldots,\rho), \tag{5} \]
has order \(\varkappa_j\) at the point \(P_j\) \((j=1,2,\ldots,m)\), and everywhere on \(\Gamma\) satisfies condition (1).

Theorem 1. For problem (1) to be solvable it is necessary and sufficient that on the surface \(R\) there exist \(\varkappa\) points \(R_1,\ldots,R_\varkappa\) (not necessarily distinct) for which the system of congruences
\[ \sum_{j=1}^{\varkappa}\int_{P_0}^{R_j}dw_\nu \equiv -\frac{1}{2\pi i}\int_{\Gamma}\ln G\,dw_\nu +\sum_{j=1}^{m}\varkappa_j\int_{P_0}^{P_j}dw_\nu \quad(\text{modulo periods}) \tag{6} \]
\[ (\nu=1,\ldots,\rho);\qquad P_0 \text{ is an arbitrarily fixed point.} \]

Necessity. Denote the solution of the problem by \(\Phi\) and its zeros by \(R_1,\ldots,R_\chi\). The function \(F=\Psi/\Phi\), where \(\Psi\) is given by formula (3), is multiplicatively multivalued with multipliers (5), meromorphic on the whole surface, having order \(\chi_j\) at the point \(P_j\) \((j=1,2,\ldots,m)\) and poles at the points \(R_1,\ldots,R_\chi\). Consequently, \(d\ln G\) is a differential of the third kind with periods
\[ \pi(B_\nu)=\int_{\Gamma}\ln G\,d\omega_\nu+2\pi i m_\nu,\qquad \pi(A_0)=2\pi i l_\nu, \]
where \(m_\nu,l_\nu\) are integers \((\nu=1,2,\ldots,\rho)\), and with poles at the points \(P_1,\ldots,P_m,R_1,\ldots,R_\chi\). From the known relation between periods \(\bigl((^2),\) Theorem 10.6\bigr) we obtain (6).

Sufficiency. To prove sufficiency, consider the differential of the third kind
\[ d\Omega=-\sum_{j=1}^{m}\chi_j d\omega_{P_jP_0}+\sum_{j=1}^{\chi}d\omega_{R_jP_0}, \]
where \(d\omega_{PQ}\) is a complex-normalized differential of the third kind with residues \(+1\) and \(-1\) at the points \(P\) and \(Q\), respectively, and the points taken as \(R_j\) \((j=1,\ldots,\chi)\) are points satisfying system (6). On the basis of (6) and the above-mentioned relation between periods \(\bigl((^2),\) Theorem 10.6\bigr), we verify that \(F_1=\exp\int d\Omega\) is a multiplicatively multivalued function with multipliers \(1/\mu_\nu\) \((\nu=1,2,\ldots,\rho)\) and order \(-\chi_j\) at the point \(P_j\) \((j=1,2,\ldots,m)\). Then, as is easy to check, \(\Phi=\Psi F_1\) will be a solution of the problem with zeros at the points \(R_1,\ldots,R_\chi\) (see, for example, \((^1),\S 44\)).

Let us note that the choice of the points \(P_j\) \((j=1,2,\ldots,m)\) and of the branch of \(\ln G\) does not affect the solvability of system (6), as is easily verified by a simple calculation.

1. Denoting
   \[ \lambda_\nu=-\frac{1}{2\pi i}\int_{\Gamma}\ln G\,d\omega_\nu +\sum_{j=1}^{m}\chi_j\int_{P_0}^{P_j}d\omega_\nu, \qquad u_\nu(P)=\int_{P_0}^{P}d\omega_\nu \quad(\nu=1,2,\ldots,\rho), \tag{7} \]
   we rewrite system (6) in the form
   \[ \sum_{j=1}^{\chi}u_\nu(R_j)\equiv \lambda_\nu \quad \text{(modulo periods)} \tag{8} \]
   \[ (\nu=1,2,\ldots,\rho). \]

The problem of finding points \(R_1,\ldots,R_\chi\) satisfying system (8) for a given right-hand side is known as the Jacobi inversion problem. It was first applied to the solution of the Riemann problem in the work \((^5)\). For \(\chi\ge \rho\) the inversion problem is always solvable, which agrees with the results of \((^{4-6})\).

To study the question of solvability of system (8) for \(0\le \chi<\rho\), we turn to the Riemann theta function of \(\rho\) variables \(u_1(P),u_2(P),\ldots,u_\rho(P)\), denoting it briefly by \(\vartheta(u_1,\ldots,u_\rho)=\vartheta(u_\nu)\). We shall need the following properties of it \((^3)\):

I. \(\vartheta\) is an even function.

II.
\[ \vartheta\left(\sum_{j=1}^{\rho-1}u_\nu(Q_j)+k_\nu\right)=0 \]
for any divisor \(Q_1,Q_2,\ldots,Q_{\rho-1}\), where \(k_\nu\) \((\nu=1,2,\ldots,\rho)\) is a system of numbers depending only on the surface.

III. Whatever the numbers \(e_\nu\) \((\nu=1,2,\ldots,\rho)\), there exists a divisor \(Q_1^0,Q_2^0,\ldots,Q_\rho^0\) such that
\[ \vartheta\left(\sum_{j=1}^{\rho}u_\nu(Q_j^0)+e_\nu+k_\nu\right)\ne 0. \]

IV. If \(\vartheta\bigl(u_\nu(P)-e_\nu\bigr)\not\equiv 0\) as a function of the point \(P\), then it has \(\rho\) zeros on the surface \(R\), and if these are denoted by \(Q_1',Q_2',\ldots,Q_\rho'\),

the following system of congruences will hold:

\[ e_\nu \equiv \sum_{j=1}^{\rho} u_\nu\left(Q'_j\right)+k_\nu \quad \text{(modulo periods)} \]

\[ (\nu=1,\ldots,\rho). \]

Theorem 2. For the solvability of system (8) it is necessary and sufficient that

\[ \vartheta\left(\sum_{j=1}^{\rho-\chi-1} u_\nu(Q_j)+\lambda_\nu+k_\nu\right)=0 \tag{9} \]

for any divisor \(Q_1,\ldots,Q_{\rho-\chi-1}\).

Necessity follows from property II.

Sufficiency. To prove sufficiency, denote by \(l\) the greatest integer for which

\[ \vartheta\left(\sum_{j=1}^{\rho-\chi+l} u_\nu(Q_j)+\lambda_\nu+k_\nu\right)=0 \tag{10} \]

for any divisor \(Q_1,\ldots,Q_{\rho-\chi+l}\). In view of (9) and property III, such a number exists and \(-1 \le l \le \chi-1\). Let \(Q^0_1,\ldots,Q^0_{\rho-\chi+l+1}\) be a divisor for which

\[ \vartheta\left(\sum_{j=1}^{\rho-\chi+l+1} u_\nu(Q^0_j)+\lambda_\nu+k_\nu\right)\ne 0 \]

and, consequently, the function

\[ \vartheta\left(u_\nu(P)-\sum_{j=1}^{\rho-\chi+l+1}u_\nu(Q^0_j)-\lambda_\nu-k_\nu\right) \tag{11} \]

is not identically equal to zero. But, in view of (10),

\[ \vartheta\left(u_\nu(Q^0_i)-\sum_{j=1}^{\rho-\chi+l+1}u_\nu(Q^0_j)-\lambda_\nu-k_\nu\right)=0 \quad (j=1,\ldots,\rho-\chi+l+1) \tag{12} \]

and, using property IV, we are convinced of the validity of system (8), if the zeros of the function (11), distinct from \(Q_1,\ldots,Q_{\rho-\chi+l+1}\), are taken as the points \(R_1,\ldots,R_{\chi-l+1}\), and \(R_{\chi-l}=R_{\chi-l+1}=\cdots=R_\chi=P_0\).

As an application of Theorem 2, we point out that equality (9) gives a necessary and sufficient condition for the solvability of Hilbert’s problem \(\operatorname{Re}[(a-ib)F]=0\) on a finite Riemann surface, in particular on a plane multiply connected domain, if it is reduced by the known method (4) to problem (1) on the doubled Riemann surface.

If, for \(\chi<0\), as a solution of problem (1) one allows functions having \(|\chi|\) poles on the surface, then the condition for solvability of the problem will likewise be given by Theorem 2.

Perm State University
named after A. M. Gorky

Received
20 V 1963

CITED LITERATURE

¹ F. D. Gakhov, Boundary Value Problems, Moscow, 1958.
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⁴ Yu. L. Rodin, DAN, 129, No. 6 (1959).
⁵ L. I. Chibrikova, The Riemann Boundary Value Problem for Automorphic Functions, Dissertation, Kazan, 1961.
⁶ W. Koppelman, Comm. Pure and Appl. Math., 12, No. 1, 13 (1959).
⁷ H. Behnke, K. Stein, Math. Ann., 120, 430 (1949).